On the wellposedness of the KdV/KdV2 equations and their frequency maps

In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on Hs the KdV2 equation is C0-wellposed if s⩾0 and illposed (in a strong sense) if s<0.